Science:Math Exam Resources/Courses/MATH257/December 2011/Question 03 (b)

MATH257 December 2011

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Question 03 (b)
(b) According to the Fourier Convergence Theorem, for which values of $x$ in the interval should each of these series converge to $x$? Verify your conclusions for the Fourier sine series at $x=0$ and $x=1$.

Make sure you understand the problem fully: What is the question asking you to do? Are there specific conditions or constraints that you should take note of? How will you know if your answer is correct from your work only? Can you rephrase the question in your own words in a way that makes sense to you?

If you are stuck, check the hints below. Read the first one and consider it for a while. Does it give you a new idea on how to approach the problem? If so, try it! If after a while you are still stuck, go for the next hint.

Hint 1
What does the Fourier convergence theorem say? If you are not sure, you can start by look it up. It can be found in your textbook (Boyce & DiPrima). It probably is in your class notes as well.

Hint 2
It is a good idea to sketch the odd and even extensions of the function $f(x)$. Do you know why?

Hint 3
The reason we want to sketch both the odd and even extensions of $f(x)$ is because the Fourier sine series comes from the odd extension of $f(x)$, while the Fourier cosine series comes from the even extension of $f(x)$.

Checking a solution serves two purposes: helping you if, after having used all the hints, you still are stuck on the problem; or if you have solved the problem and would like to check your work. If you are stuck on a problem: Read the solution slowly and as soon as you feel you could finish the problem on your own, hide it and work on the problem. Come back later to the solution if you are stuck or if you want to check your work. If you want to check your work: Don't only focus on the answer, problems are mostly marked for the work you do, make sure you understand all the steps that were required to complete the problem and see if you made mistakes or forgot some aspects. Your goal is to check that your mental process was correct, not only the result.

Solution

Found a typo? Is this solution unclear? Let us know here. Please rate my easiness! It's quick and helps everyone guide their studies. In short, the Fourier convergence theorem states the following. If $f$ is a periodic function such that $f$ and $\displaystyle {}f'$ are piecewise continuous, then the Fouriner series converges to $f(x)$ at all points where $f$ is continuous and converges to ${\frac {1}{2}}[f(x+)+f(x-)]$ at all points where $f$ is discontinuous. In the above, $f(x+)$ denotes the right hand limit of $f$ at $x$, while $f(x-)$ denotes the left hand limit of $f$ at $x$. In other words, for a real number $a$, $f(a+)=\lim _{x\to a^{+}}f(x)\;\;{\text{and}}\;\;f(a-)=\lim _{x\to a^{-}}f(x).$ With the Fourier convergence theorem in mind, let's find out the point of convergence of the Fourier cosine series $Cf(x)$ at each point $x$ in the interval $[0,1]$. To do so, we first need to understand the origin of the Fourier cosine series. Recall that the Fourier cosine series is the Fourier series of $f_{\text{even}}$ where $f_{\text{even}}$ is the even extension of $f$. Following the Fourier convergence theorem, to find the point of convergence of the Fourier cosine series, we need to know where the points of continuity and discontinuity of $f_{\text{even}}$ are. We will do so by sketching the graph of $f_{\text{even}}$. To begin, we sketch the graph of $f(x)$. A sketch of $f(x)$ The graph of $f_{\text{even}}$ is obtained by first reflecting $f$ over the y-axis and hence obtaining a function over the interval $[-1,1]$. We then periodic extend this function to get $f_{\text{even}}$ which will have a period of $2$. A sketch of the even extension of $f(x)$. The part of the function from $-1$ to $1$ is coloured in red. The red part shows one period of $f_{\text{even}}$. From the sketch of $f_{\text{even}}$, we see that $f_{\text{even}}$ is continuous everywhere. Hence, by the Fourier convergence theorem, the Fourier cosine series $Cf(x)$ converges to $f_{\text{even}}$ for every real number $x$. Now, since $f$ agrees with $f_{\text{even}}$ at every point in the interval $[0,1]$, $Cf(x)$ converges to $f(x)=x$ at every point $x$ in the interval $[0,1]$. Similarly, let $f_{\text{odd}}$ be the odd extension of $f$. Then the Fourier sine series $Sf(x)$ is the Fourier series of $f_{\text{odd}}$. We will sketch $f_{\text{odd}}$ to find its points of continuity and discontinuity. The graph of $f_{\text{odd}}$ is obtained by first rotating $f$ about the origin by 180 degree and hence obtaining a function over the interval $[-1,1]$. We then periodic extend this function to get $f_{\text{odd}}$ which will have a period of $2$. A sketch of the odd extension of $f(x)$. The part of the function from $-1$ to $1$ is coloured in red. The red part shows one period of $f_{\text{odd}}$.The values at points of discontinuity is not specified in the sketch. From the sketch of $f_{\text{odd}}$ is discontinuous at points $x=-1+2k$ with k an integer. Hence, by the Fourier convergence theorem, the Fourier sine series $Sf$ converges to $f_{\text{odd}}$ at every point in the interval $[0,1{\color {red})}$, but at $x=1$, $Sf$ converges to ${\frac {1}{2}}[f_{\text{odd}}(1-)+f_{\text{odd}}(1+)]={\frac {1}{2}}[1+(-1)]=0$. Since $f_{\text{odd}}$ agrees with $f$ in the interval $[0,1)$, $Sf(x)$ converges to $f(x)=x$ at every point in the interval $[0,1{\color {red})}$ and at $x=1$, $Sf(1)$ converges to 0. Finally, the question asks us to verify our conclusions for the Fourier sine series at x = 0 and x = 1. Evaluate the sine series found at part a at $x=0$, we get $Sf(0)=\sum _{n=1}^{\infty }{\frac {2}{n\pi }}(-1)^{n+1}\underbrace {\sin(n\pi \;0)} _{=0}=\sum _{n=1}^{\infty }0=0$ and evaluating at $x=1$, we get $Sf(1)=\sum _{n=1}^{\infty }{\frac {2}{n\pi }}(-1)^{n+1}\underbrace {\sin(n\pi )} _{=0}=\sum _{n=1}^{\infty }0=0$ as expected.